The tight upper-bound conjecture for generalized Ramsey numbers of even cycles

Let $r(K_{n,n},C_{2k},q)$ be the generalized Ramsey number for edge-colorings of $K_{n,n}$, and consider the case $q=3$. For $k\geq2$, the conjectured asymptotic value is

$$r(K_{n,n},C_{2k},3)=\frac{3k-2}{2(k-1)(2k-1)}n+o(n).$$

Tight upper-bound conjecture. For all $k\geq2$,

$$r(K_{n,n},C_{2k},3)=\frac{3k-2}{2(k-1)(2k-1)}n+o(n).$$

This conjecture asserts that the paper's upper bound is asymptotically tight for every $k\geq2$. The case $k=3$ is proved in the paper, whereas tightness for the other values remains open.